Refutation of Products of Linear Polynomials
|
|
- Lesley Day
- 5 years ago
- Views:
Transcription
1 Refutation of Products of Linear Polynomials Jan Horáček and Martin Kreuzer Faculty of Informatics and Mathematics University of Passau, D Passau, Germany Abstract. In this paper we consider formulas that are conjunctions of linear clauses, i.e., of linear equations. Such formulas are very interesting because they encode CNF constraints that are typically very hard for SAT solvers. We introduce a new proof system SRES that works with linear clauses and show that SRES is implicationally and refutationally complete. Algebraically speaking, linear clauses correspond to products of linear polynomials over a ring of Boolean polynomials. That is why SRES can certify if a product of linear polynomials lies in the ideal generated by some other such products, i.e., the SRES calculus decides the ideal membership problem. Furthermore, an algorithm for certifying inconsistent systems of the above shape is described. We also establish the connection with an another combined proof system R(lin). Keywords: linear clauses, Boolean polynomials, algebraic proof systems, SAT solving 1 Introduction In this paper we deal with a special case of the ideal membership problem: given a set of Boolean polynomials S which are products of linear polynomials and a Boolean polynomial f which is also a product of linear polynomials, decide whether f S. Traditionally, one can use Gröbner bases to solve this problem. The assumption that the given polynomials are products of linear polynomials allows us introduce a new calculus, called SRES, that is tailored to tackle this problem. It needs only a lightweight version of the S -polynomials used Gröbner basis algorithms, namely the s-resolvents we introduced in [6]. These s-resolvents also generalize the classical resolution rule from propositional logic. Nevertheless, we consider SRES to be an algebraic proof system (in the sense of [5], [2] or [4]), and hence we mostly use the terminology of commutative algebra to describe it. The main difference in the algebraic approach is that we study assignments that yield False, i.e., are zeros of a system, in constrast to look for assignments that yield True in the SAT terminology. From the propositional logic point of view, we would like to decide if the semantic implication S = f holds. Recall that a linear Boolean polynomial x i1 + + x ij corresponds to a linear XOR constraint x i1 x ij. Such constraints are very difficult for SAT solvers because the truth value depends 33
2 34 Jan Horáček and Martin Kreuzer genuinely on all variables, i.e., changing the value of one variable results in flipping the truth value of the XOR. Many hard formulas in CNF can be constructed by combining literals into linear XOR constraints, resulting in so-called linear clauses. For details, we refer the readers to [1] or [12]. Such formulas appear quite frequently in cryptography, where linear constraints are typically mixed with highly nonlinear ones, for instance in substitution permutation networks. Another application is to handle the bit-blasted version of ARX operations (Add- Roll-Xor) which is becoming a popular part of ARX ciphers. When converted to bit-wise operations, these conversions give XOR-heavy formulae with a handful of AND gates describing the carry chains of the adders. The paper is organized as follows. In Section 2 we recall the definition of linear clauses and introduce several ways of describing them, in particular using products of linear Boolean polynomials. Our main data structure to describe them is the set of linear polynomials that appear in the linear clause. In Section 3 we define the proof system SRES which incorporates and extends the s-resoution rule defined in [6]. We prove that the SRES proof system is implicationally and refutationally complete. Moreover, we establish a relation with other combined systems that use resolution and polynomial calculus (see [11]). In particular, we show that SRES simulates the proof system R(lin) defined in [7, 11]. In Section 4 we describe a concrete algorithm that searches for SRES-refutation proofs of inconsistent systems, called the SRES Refutation Algorithm, and compare it to other approaches to the problem at hand. Finally, in Section 5, we apply the SRES Refutation Algorithm to some examples and point out possible further improvements. In the following we use some of the definitions and results given in [6], in particular the algorithm Sres given there. However, to aid the readers, we tried to make the paper as self-contained as possible and mention any overlaps explicitly. 2 Background Let F 2 = Z/2Z be the binary field and F 2 [x 1,..., x n ] a polynomial ring over F 2. The ring B n = F 2 [x 1,..., x n ]/F with F = x x 1,..., x 2 n + x n is called the ring of Boolean polynomials in the indeterminates x 1,..., x n. Let L n be the set of all linear polynomials in F 2 [x 1,..., x n ], i.e., the set of all polynomials of degree 1. (Here we use deg(0) = 1.) Throughout the paper we consider the obvious correspondences between the following types of objects, where l 1,..., l k L n : (C1) A set of linear polynomials H = {l 1,..., l k } L n. (C2) A Boolean polynomial h = k i=1 l i which is a product of linear polynomials in B n. (C3) A linear clause (l 1 = 0) (l k = 0). Next we give an example of the above correspondence.
3 Refutation of Products of Linear Polynomials 35 Example 1 Let H = {x 1 + x 2 + 1, x 1 + x 3 } L 3. Then h = (x 1 + x 2 + 1)(x 1 + x 3 ) = x 1 x 3 + x 1 x 2 + x 2 x 3 + x 3, and H (resp. h) coresponds to the propositional logic formula (x 1 x 2 1 = 0) (x 1 x 3 = 0). To simplify the notation, we view the sets in (C1) as polynomials in (C2), e.g., we speak of zeros of H instead of zeros of h. Furthermore, we always assume that the linear polynomials l 1,..., l k appearing in (C1) are pairwise distinct, i.e., that the set H does not contain duplicates. The notion of linear clauses has been considered before in [7,11]. Note that many difficult CNF instances can be naturally encoded in the form (C3). Thus we are targeting very hard formulas (see [10, Sec. 3]). Furthermore, we observe that two different subsets in (C1) may represent the same polynomial, as the following example shows. Example 2 Consider the sets of linear polynomials {x 3, x 2 + x 3, x 1 + x 3 + 1} and {x 3, x 1 +x 2, x 1 +x 3 +1} as in (C1). Then we have x 3 (x 2 +x 3 )(x 1 +x 3 +1) = x 3 (x 1 + x 2 )(x 1 + x 3 + 1) in B 3, i.e. the corresponding polynomials in (C2) agree. The subsets in (C1) are measured by degree and size. We recall here some definitions from [6] and [11]. Definition 3 Let H = {l 1,..., l k } L n. (1) The number deg(h) = #H is called the degree of H. (2) For l L n, we let var(l) be the set of indeterminates occurring in l. We call the number size(h) = # var(l 1 ) + + # var(l k ) the size of H. Our goal in this paper is to show that a given set of linear clauses is unsatisfiable. For this purpose, we define semantic implication as follows. Definition 4 Let F 1,..., F m, H L n. We say that F 1,..., F m semantically imply H if every F 2 -rational common zero of the Boolean polynomials corresponding to F 1,..., F m is a zero of H. In this case we write F 1,..., F m = H. From the algebraic point of view, we have F 1,..., F m = H if and only if H F 1,..., F m B n. Note that the zeros of an ideal in B n are always F 2 - rational, because ideals in B n correspond to ideals in F 2 [x 1,..., x n ] that contain the field ideal x x 1,..., x 2 n + x n. Using the newly defined terminology, we can say that our goal is to refute a given set of linear clauses, i.e. to prove that the corresponding sets of linear polynomials semantically imply {1}. 3 The SRES Proof System Recall that a proof system, sometimes called a Hilbert system or Hilbert calculus, consists of a syntax, i.e. a set of rules which determine the set of well-formed formulas of the system, by a set of axioms, i.e. a set of formulas which are assumed to be tautologies, and by its set of rules of inference, i.e. by a set of rules which determine how one can get new tautologies from known ones.
4 36 Jan Horáček and Martin Kreuzer For instance, the Gröbner proof system defined in [5] admits all formulas of the form f(x 1,..., x n ) = 0 where f F 2 [x 1,..., x n ]. Its axioms are the Boolean axioms x 2 i + x i = 0 for i = 1,..., n, and its rules of inference are f f + g g (P a ) and f x i f (P m) where f, g F 2 [x 1,..., x n ] and x i is an indeterminate. In this section we continue to study the proof system based on s-resolution which was introduced in [6]. More precisely, we extend that system by other inference rules, and thus we define the new proof system called SRES. Let us begin by recalling the proof system in [6]. Definition 5 The proof system s-resolve is defined by the following parts: (1) A formula is a set of linear polynomials {l 1,..., l s } L n, where s N +. (2) The axioms are the Boolean axioms given by {x i, x i + 1} for i = 1,..., n. (3) There exists one rule of inference (R), namely s i=1 {l i} G s i=1 {l i + 1} G s 1 i=1 {l i + l i+1 + 1} G G (R) where G, G L n and s 1. This rule is also called s-resolution. (For s = 1, we let s 1 i=1 {l i + l i+1 + 1} =.) The result of an application of the s-resolution rule is called an s-resolvent. Let us note that the Boolean axioms immediately imply the following remark. Remark 6 By applying 2-resolution to the axioms {x i, x i +1} and {x i +1, x i }, we obtain that {0} is a tautology in the s-resolve proof system. In [6] we showed that s-resolve is correct in the sense of the following definition. Definition 7 Let (I) be a rule of inference of a proof system. We say that the rule of inference (I) is correct if F 1 F 2... F m H (I) for some formulas F 1,..., F m, H implies F 1,..., F m = H. For s 3, the s-resolution rule depends e.g. on the numbering of the linear polynomials. (2-resolvents are unique because there is only one way how to form the linear polynomial l 1 +l 2 +1.) However, considered as a Boolean polynomial, the s-resolvent is uniquely determined. The next example is a point in case.
5 Refutation of Products of Linear Polynomials 37 Example 8 Resolving two sets F = {x 1 + 1, x 1 + x 3, x 1 + x 2 + 1, x 2 + x 3 + 1} and G = {x 1, x 1 + x 3 + 1, x 1 + x 2, x 2 + x 3 } with the numbering l 1 = x 1 + 1, l 2 = x 1 + x 3, l 3 = x 1 + x 2 + 1, l 4 = x 2 + x yields the 4-resolvent R 1 = {x 3, x 2 + x 3, x 1 + x 3 + 1}. If we swap the last two polynomials in G, 4-resolution with the numbering l 1 = x 1 + 1, l 2 = x 1 + x 3, l 4 = x 1 + x 2 + 1, l 3 = x 2 + x yields R 2 = {x 3, x 1 + x 2, x 1 + x 3 + 1}. Both R 1 and R 2 correspond to the same Boolean polynomial, as we saw in Example 2. Next we enrich s-resolve with a new rule of inference as follows. Definition 9 The proof system SRES is defined by the following parts. (1) The syntax agrees with the syntax of s-resolve, i.e. a formula is a set of linear polynomials {l 1,..., l s } L n, where s N +. (2) The axioms are the Boolean axioms {x i, x i + 1} for i = 1,..., n. (3) The rules of inference consist of s-resolution (R) and the following weakening rule (W): H (W) H {l} for H L n and l L n. Since we trivially have H = H {l}, the proof system SRES is correct with respect to semantic implication. Next we recall the definition of a proof. Definition 10 Let PS be a proof system. A PS-proof of a formula H from the initial premises F 1,..., F m in the proof system PS is a sequence of formulas π = (G 1,..., G k ) such that G k = H and each of the formulas G i is of one of the following forms: (1) G i {F 1,..., F m } (2) G i is one of the axioms of PS. (3) G i is obtained from some of the formulas G j with j < i by applying one of the rules of inference of the proof system PS. If a formula H has a PS-proof from {F 1,..., F m }, we write F 1,..., F m PS H, or simply F 1,..., F m H if no confusion can arise. Since the proof system SRES is correct, our goal to refute {F 1,..., F m } can now be expressed by asking that we should prove F 1,..., F m {1}. Notice that the intended semantics is that an unsatisfiable formula corresponds to polynomial equations l i,1 l i,s = 0 for i {1,..., m} which has no solution. Equivalently, a tautology is an equation which holds for all points of F n 2. A proof can be seen as a sequence of applications of rules to axioms or previously proved tautologies. Next we extend the capabilities of our proof system by deriving the following further rules of inference. Definition 11 Let F, G be subsets of L n, and let l, l 1, l 2 L n.
6 38 Jan Horáček and Martin Kreuzer (1) The rule (U) defined by is called unit cancellation. (2) The rule (MP) defined by H {1} H (U) {l} H {l + 1} H (MP) is called modus ponens. (3) The rule (A) defined by F {l 1 } H {l 2 } F H {l 1 + l 2 } (A) is called the addition rule. Proposition 12 The rules (U), (MP), and (A) are correct. Furthermore, any proof derived using (U), (MP) and (A) can be rewritten into an SRES-proof. Proof. The correctness of (U) follows from the observation that multiplying a Boolean polynomial by 1 does not change its zeros. Using Remark 6 we apply 1-resolution on H {1} and {0}, and we get H. To prove modus ponens it suffices to use (R) with s = 1 and G =. Finally, we show the correctness of (A). Using the weakening rule, we infer F {l 1, l 2 + 1} from F {l 1 } and H {l 2, l 1 +1} from H {l 2 }. Then, using 2-resolution, we get F H {l 1 +l 2 }. The following remark and the subsequent proposition will become important later in the proof of Proposition 18. Remark 13 Let π = (H 1,..., H k ) be an SRES-proof of H k from F 1,..., F m that uses only the s-resolution rule. Let l be an element of one of the sets H i. Then l lies in the F 2 -vector space generated by the linear polynomials contained in the union m i=1 F i. Proposition 14 Let F L n and let i {1,..., n}. For a {0, 1}, let F (x i a) denote the set which is obtained by substituting x i a into the linear polynomials contained in F. Then the following claims hold true for i {1,..., n}. (1) F, {x i } SRES F (x i 0) (2) F (x i 0), {x i } SRES F (3) F, {x i + 1} SRES F (x i 1) (4) F (x i 1), {x i + 1} SRES F
7 Refutation of Products of Linear Polynomials 39 Proof. First we prove (1). Let l F be such that x i occurs in l. The polynomial x i + l corresponds to substituting x i = 0 into l. This addition can be derived in SRES using Proposition 12. The claims (2) (4) follow analogously from Proposition 12. The next example illustrates this proposition. Example 15 Let F 1 = {x 1 + x 2, x 1 } and F 2 = {x 1 + 1}. By Proposition 12, there exists an SRES-proof of the set {x 2 + 1, 1} from F 1, F 2 which corresponds to the substitution of x 1 = 1 into F 1. On the other hand, we can backtrack the substitution, i.e. we have F 2, {x 2 + 1, 1} SRES F 1. The following example shows an important advantage of the SRES proof system over the Gröbner proof system defined in [5]. More precisely, it shows that the data structures (C1) (C3) efficiently store dense linear clauses. Example 16 Consider the sets F 1 = {x 1 + x 2,..., x n + x n+1 } F 2 = {x 1 + x 2 + 1} F 3 = {x 2 + x 3 + 1}. =. F n+1 = {x n + x n+1 + 1} On one hand, it is easy to see that the system is inconsistent because substituting F 2,..., F n+1 into F 1 gives us 1. On the other hand, the input for the Gröbner basis algorithm is assumed to be expanded (i.e., not in the form of products of linear polynomials). We can always find n N such that expanding F 1 to the polynom f 1 = n i=1 (x i + x i+1 ), which has 2 n terms, exceeds the available memory, and hence any Gröbner basis algorithm can not be applied. Notice that we have size(f 1 ) = 2n. One workaround would be to introduce new indeterminates to break the long product, but it does not help too much because the Gröbner basis algorithm substitutes the new indeterminates back, and thus recovers the expanded polynomial F 1 again. However, by Proposition 14, there exists a short SRES refutation that corresponds to the substitution of F 2,..., F n+1 into F 1. The following definition provides further useful properties of proof systems. Definition 17 (1) A proof system is called implicationally complete if for every formula H and every set of formulas {F 1,..., F m } such that we have F 1,..., F m = H, there exists a proof of H from F 1,..., F m in this proof system. (2) A proof system is called refutationally complete if for every inconsistent set of formulas {F 1,..., F m }, i.e. for F 1,..., F m = {1}, there exists a proof of {1} from F 1,..., F m in this proof system.
8 40 Jan Horáček and Martin Kreuzer The proof of the following proposition is inspired by the corresponding proof for the classical resolution calculus (see for instance [3, Th ]) and by the proof given in [11, Th. 5.1]) Proposition 18 The proof system SRES is implicationally and refutationally complete. Proof. First we show that SRES is implicationally complete. Let F 1,..., F m, H L n be sets of linear polynomials such that F 1,..., F m = H. We want to prove that F 1,..., F m SRES H. We proceed by induction on the number k = size(f 1 ) + + size(f m ) + size(h). Let us consider the case k = 0, i.e. the case when all linear polynomials in F i and H are constants 0 or 1. If all sets F 1,..., F m are equal to {0}, there is trivially an SRES-proof of {0}. All the semantic implicants of {0} of size 0 can be derived from {0} by the weakening rule. If there exists a set F i = {1}, then there is trivially an SRES-proof that refutes F 1,..., F m, and hence we can derive any linear clause by the weakening rule and the unit cancellation rule. Finally, if F i = {0, 1}, then F i is simplified to {0} by unit cancellation, and thus we can use the previous argument. Now assume that the claim holds for some k 0 and consider a semantic implication F 1,..., F m = H in which the sum of sizes of F 1,..., F m, H is at most k+1. Choose i {1,..., n} such that x i occurs in F 1 F m H. Given a {0, 1}, let F (x i a) denote the set which is obtained by substituting x i a into the linear polynomials contained in F. By Proposition 14, we have F j, {x i } SRES F j (x i 0) for j = 1,..., m. Moreover, we clearly have F 1 (x i 0),..., F m (x i 0) = H(x i 0). By the induction hypothesis, there is an SRES-proof of H(x i 0) from F 1 (x i 0),..., F m (x i 0). Altogether, we get {x i }, F 1,..., F m SRES H(x i 0). Furthermore, by Proposition 14, we have H(x i 0), {x i } SRES H. All in all, there is an SRES-proof π 1 of H from {x i }, F 1,..., F m. Analogously, we get an SRES-proof π 2 of H from {x i + 1}, F 1,..., F m. Next we modify the derivations of π 1 (resp. π 2 ) such that they start from {x i, x i + 1}, F 1,..., F m instead of {x i }, F 1,..., F m (resp. {x i + 1}, F 1,..., F m ). Note that the same rules of inference can be applied, and thus one can rewrite the proof π 1 into an SRES-proof α 1 from {x i, x i + 1}, F 1,..., F m of either H or H {x i }. Similarly, we rewrite π 2 into an SRES-proof α 2 from {x i, x i + 1}, F 1,..., F m of H or H {x i + 1}. If the proof α 1 or α 2 ends with H, we are done. Otherwise, we have H {x i }, H {x i + 1} SRES H by a single step of the 1-resolution rule, which concludes the proof. The SRES proof system is in fact a combined proof system in the sense of [8, Sec.7.1]. For instance, R(lin) in [11] is an another example of a combined proof
9 Refutation of Products of Linear Polynomials 41 systems that combines resolution with polynomial calculus. By Proposition 12 we immediately get that SRES efficiently simulates the system R(lin). This means that there exists a polynomial-time algorithm that translates any R(lin)-proof of H from F 1,..., F m to an SRES-proof of H from F 1,..., F m. On the other hand, SRES cannot simulate the rule (P a ) of the Gröbner proof system because addition of two products of linear polynomials does not have to be a product of linear polynomials in B n, e.g., x 1 x cannot be written as a product of linear polynomials in B 3. 4 The SRES Refutation Algorithm In [6] the authors introduced an algorithm that finds refutation proofs for unsatisfiable formulas in CNF using s-resolution. In this section we generalize this result to formulas that are conjuctions of linear clauses and show that it is refutationally complete in the sense of Definition 17. We recall the following algorithm for computing the s-resolvent of two polynomials which is described in [6]. Algorithm 1 Sres ( s-resolution of Two Polynomials) Input: Sets F, G L n. We assume that #{l F l + 1 G} 1. Output: A set R L n such that R is the s-resolvent of F and G. 1: Write {l F l + 1 G} as {l 1,..., l s}. 2: F := {l F l + 1 / G} 3: G := {l G l + 1 / F } 4: R := F G 5: if s = 1 and R = then 6: return {1} 7: else 8: for i = 1,..., s 1 do 9: if l i + l i+1 / R then 10: R := R {l i + l i+1 + 1} 11: else 12: return {0} 13: end if 14: end for 15: end if 16: return R Algorithm 2 extends two polynomials by the weakening rule in all possible ways such that s-resolution can be applied. More precisely, this set of extensions is defined as follows. Definition 19 Let F, G L n. The set of all pairs K L n L n such that the following two conditions hold for all (F, G ) K (1) #{l F l + 1 G } 1
10 42 Jan Horáček and Martin Kreuzer (2) F G F G {l + 1 l F G} is called the set of expansions for F, G. Condition (1) encodes the fact that there exists at least one l L n in F and G on which we can s-resolve. Condition (2) restrains F, G to contain linear polynomials l or l + 1 such that l F G. Note that the set of expansions for F, G is unique. Algorithm 2 produces the set of expansions in a brute-force way, i.e., Condition (1) is implemented in Step 10, and Condition (2) is fulfilled because of the two foreach loops in Steps 3, 4. Algorithm 2 AllExpansions (All Possible Expansions to s-resolution) Input: Sets F, G L n. Output: The set of expansions for F, G. 1: {l 1,..., l k } := {l F l / G, l + 1 / G} 2: {l 1,..., l k } := {l G l / F, l + 1 / F } 3: foreach A {l 1, l 1 + 1, 1} {l k, l k + 1, 1} do 4: foreach B {l 1, l 1 + 1, 1} {l k, l k + 1, 1} do 5: Write A = (a 1,..., a k ). 6: Write B = (b 1,..., b k). 7: F := F k i=1 {ai} 8: G := G k i=1 {bi} 9: Minimize the representation of F and G by applying unit cancellation, i.e., remove the element 1 from F, G. 10: if #{l F l + 1 G } 1 then 11: K := K {(F, G )} 12: end if 13: end foreach 14: end foreach 15: return K The next example shows the generation of the pairs in Algorithm 2. Example 20 Let F = {x 1, x 2, x 3 + 1} and G = {x 1 + 1, x 2, x 4 }. Then we may extend F to F itself, to {x 1, x 2, x 3 + 1, x 4 }, or to {x 1, x 2, x 3 + 1, x 4 + 1}. Similarly, we may extend G to G, to {x 1 + 1, x 2, x 4, x 3 + 1}, or to {x 1 + 1, x 2, x 4, x 3 }. Altogether, nine pairs are constructed. Next we process the pairs computed by Algorithm 2 in increasing order with respect to the following ordering relation. Definition 21 Let F, G, F 1, F 2, G 1, G 2 L n. (1) We write F G if we have deg(f ) < deg(g), or if we have deg(f ) = deg(g) and size(f ) < size(g).
11 Refutation of Products of Linear Polynomials 43 (2) Let #{l F 1 l + 1 G 1 } 1 and #{l F 2 l + 1 G 2 } 1. We write (F 1, G 1 ) (F 2, G 2 ) if Sres(F 1, G 1 ) Sres(F 2, G 2 ). Note that the size of the s-resultants can be determined without actually executing Sres (see [6, Alg. 8]). Now we are ready to present Algorithm 3 for constructing SRES-refutations. We assume that all sets occurring in the algorithm do not contain duplicates, i.e., that removal of duplicates is applied whenever possible. This is the classical assumption on sets in programming languages such as python. Algorithm 3 SRES Refute (SRES Refutation Algorithm) Input: Subsets F 1,..., F m L n such that F i does not contain any duplicate elements. Output: False if F 1,..., F m = {1}, True otherwise. Require: Algorithms 1, 2. 1: S := {F 1,..., F m} 2: if {1} S then 3: return False 4: end if 5: Apply unit simplification and cancellation on S. 6: {Q 1,..., Q k } := S n i=1 {xi, xi + 1} 7: Let P be the list containing all pairs computed by AllExpansions(Q i, Q j) for 1 i < j k. 8: while P do 9: Let (F, G) be a minimum of P w.r.t., and remove (F, G) from P. 10: R :=Sres(F, G) 11: if R = {1} then 12: return False 13: else if R is not a subset of any Q S then 14: Remove all Q from S with R Q and all pairs (Q 1, Q 2) from P such that R Q 1 or R Q 2. 15: Append all pairs in AllExpansions(R, G) for G S, R G to the list P. 16: S := S {R} 17: end if 18: end while 19: return True Proposition 22 Algorithm 3 is correct, refutationally complete, and finite. In particular, the algorithm returns False if and only if F 1,..., F m = {1}. Proof. Finiteness follows from the fact that there are only finitely many linear polynomials in L n. Hence the sets in S are finite, and so is the set P L n L n. The algorithm is correct because the SRES inference rules semantically imply their results. It remains to show that the algorithm is refutationally complete. Assume that the ideal generated by the polynomials corresponding to the input sets has no
12 44 Jan Horáček and Martin Kreuzer common zero. By Proposition 18, we know that there exists an SRES-refutation of F 1,..., F m. The Boolean axiom is incorporated in Step 6, and weakening takes place in the function AllExpansions such that all possible choices to form an s-resolvent are created for all possible s N +. The s-resolution rule is then applied by calling Sres. Any set Q that is a proper superset of some set already occurring in S is ignored because all possible s-resolvents that would be created using Q are at that time already in P. Thus the algorithm sequentially creates all SRES-derivations from F 1,..., F m. It arranges the computation such that smaller sets in terms of size are preferred. Since Proposition 18 shows that there exists an SRES-refutation, the set {1} is eventually discovered by the algorithm. The list P in Algorithm 3 can be implemented as a min-heap such that the minimal pair is always easily found and extracted. The number of pairs in P may be huge. Thus it is convenient to compute the s-resolvents only in Step 10. The next example indicates how the pairs can be stored. The s-resolvent is created only if its size is minimal. Example 23 Let F 1 = {x 1 +1} and F 2 = {x 1, x 2 }. Algorithm AllExpansions outputs (F 1, F 2 ), (F 1 {x 2 + 1}, F 2 ), (F 1 {x 2 }, F 2 ). The pair (F 1 {x 2 }, F 2 ) can be stored as a tuple (1, {x 2 }, 2,, 1) with the meaning Sres of F 1 {x 2 } and F 2 has size 1. Recall that we need to know the sizes of all s-resultants in P in order to select the minimum. Thus the chosen format is very convenient because the size of the s-resolvent can be predicted as in [6, Alg. 8] without computing the actual s-resolvents. Furthermore, one can form only 2-resolvents in Algorithm 3 since 2-resolution is enough to simulate R(lin) which is implicationally and refutationally complete. However, extra linear clauses coming from s-resolution steps with s 3 may come in handy, and the refutation can be found faster in Algorithm 3. 5 Examples and Future Directions In this section we give some examples of the SRES proof system. Initial experiments on refuting CNF formulae using SRES can be found in [6, Sec. 9]. Those experiments were focused on comparing resolution with SRES based on CNF benchmarks coming from [10]. These formulae are hard for classical resolution, but there may exist short refutations in other axiomatic systems of propositional calculus [13]. In the following example taken from [5], we compare SRES with algebraic systems such as the Gröbner proof system and the Nullstellensatz system [4, Def. 2.1]. Let P = F 2 [x 1,..., x n ]. A Nullstellensatz proof of a polynomial h P from polynomials f 1,..., f m P is a tuple of polynomials (p 1,..., p m, r 1,..., r n ) P m+n such that the equation
13 Refutation of Products of Linear Polynomials 45 m p i f i + i=1 n r j (x 2 j + x j ) = h j=1 holds in P. The degree of the proof is defined as max { max i ( deg(pi ) + deg(f i ) ) (, max deg(rj ) + 2 )}. j Example 24 Consider the polynomials f 1 = x 1 + x 1 x 2 and f 2 = x 2 + x 2 x 3 in F 2 [x 1, x 2, x 3 ]. They encode two implications x 1 x 2 and x 2 x 3. (E.g., if x 1 = 1, then x 2 is constrained to be 1.) Let us write a proof of h = x 1 + x 1 x 3, i.e., the implication x 1 x 3, in the Gröbner proof system. Firstly, we multiply f 2 by x 1, and we get g 1 = x 1 x 2 + x 1 x 2 x 3. The addition f 1 + g 1 gives us x 1 + x 1 x 2 x 3. Then we compute g 2 = x 3 f 1 = x 1 x 3 + x 1 x 2 x 3. Finally, we get g 1 + g 2 = x 1 + x 1 x 3. Note that the maximal degree appearing in the proof is 3. On the other hand, there exists a Nullstellensatz proof of the maximal degree O(log(n)) of x 1 x n from x 1 x 2,..., x n 1 x n (see [5]). In our case, the Nullstellensatz proof is the tuple (1 + x 3, x 1, 0, 0, 0) because of the equality (1 + x 3 )(x 1 + x 1 x 2 ) + x 1 (x 2 + x 2 x 3 ) = x 1 + x 1 x 3. Now we write the polynomials f 1, f 2 as F 1 = {x 1, 1 + x 2 } and F 2 = {x 2, 1 + x 3 }. Resolving on x 2 yields H = {x 1, 1+x 3 } which correponds to the polynomial h. Note that the maximal degree in the proof is now 2, and the SRES proof is simpler than the Gröbner proof. Further typical examples that are easy for SRES and difficult for resolution [13] are inconsistent systems of linear equations. By Proposition 12, SRES simulates the addition rule, and thus one can use Gaußian elimination to produce SRES-refutations for such systems. Recall the encoding into linear clauses in Example 16 is very efficient for SRES, but multiplying out the products causes an exponential blowup. A similar problem emerges in the next example in the case of CNF encodings. Example 25 Consider the linear system f 1 = x x n, f 2 = x x n + 1 over F 2. On one hand, encoding f 1 and f 2 in CNF suffers from introducing either exponentially many auxiliary variables or exponentially many new clauses. On the other hand, by the weakening rule and 2-resolution we get f 1 (f 2 + 1) + (f 1 + 1)f 2 = 1 in B n immediately. Let us conclude this section and this paper with a few remarks about the problem that is solved by Algorithm 3. The traditional way in Computer Algebra is to use Gröbner basis algorithms. As we saw in Example 16, they have serious drawbacks in our setting and tend to run out of memory quickly.
14 46 Jan Horáček and Martin Kreuzer Another approach is to use a SAT solver, e.g. a version of the DPLL algorithm. However, the DPLL approach needs a huge number of clauses to describe XOR constraints. All in all, the classical DPLL algorithm can be extended to linear clauses in a rather straightforward way, as for instance done in [7], but appears to be not very efficient. The new approach introduced here, namely the SRES proof system and the SRES Refutation Algorithm, offers the prospect of a new and tailor-made way to treat systems of linear clauses. Besides finding possible optimizations of the implementation, we may enhance it further by combining it with a suitable conflict-learning mechanism such as the ones which lie at the heart of modern SAT solvers. Recall that conflict clauses in SAT solvers can be generated from 1-resolvents of a cut in the implication graph. Thus, s-resolution may generalize conflict learning in the case of linear clauses or help to improve a separate XORreasoning module such as one introduced in [9]. Acknowledgments. The authors thank Jan Krajíček and Iddo Tzameret for fruitful discussions on combined proof systems. We thank the anonymous reviewers for their many insightful comments. This work was financially supported by the DFG project Algebraische Fehlerangriffe [KR 1907/6-2]. References 1. Baumgartner, P., and Massacci, F. The taming of the (X)OR. In Computational LogicCL Springer, 2000, pp Berkholz, C. The relation between polynomial calculus, Sherali-Adams, and sum-of-squares proofs. In LIPIcs-Leibniz International Proceedings in Informatics (2018), vol. 96, Leibniz-Zentrum fuer Informatik, Schloss Dagstuhl. 3. Büning, H. K., and Lettmann, T. Propositional logic: deduction and algorithms, vol. 48. Cambridge University Press, Buss, S., Impagliazzo, R., Krajíček, J., Pudlák, P., Razborov, A. A., and Sgall, J. Proof complexity in algebraic systems and bounded depth frege systems with modular counting. Computational Complexity 6, 3 (1996), Clegg, M., Edmonds, J., and Impagliazzo, R. Using the Groebner basis algorithm to find proofs of unsatisfiability. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (New York, USA, 1996), STOC 96, ACM Press, pp Horáček, J., and Kreuzer, M. On conversions from CNF to ANF. (submitted) (2018). 7. Itsykson, D., and Sokolov, D. Lower bounds for splittings by linear combinations. In International Symposium on Mathematical Foundations of Computer Science (2014), Springer, pp Krajicek, J. Proof complexity (book draft). Available at krajicek/prfdraft2.pdf, Laitinen, T., Junttila, T., and Niemelä, I. Conflict-driven xor-clause learning. In Theory and Applications of Satisfiability Testing SAT 2012 (Berlin, Heidelberg, 2012), A. Cimatti and R. Sebastiani, Eds., Springer Berlin Heidelberg, pp
15 Refutation of Products of Linear Polynomials Lauria, M., Elffers, J., Nordström, J., and Vinyals, M. CNFgen: A generator of crafted benchmarks. In Theory and Applications of Satisfiability Testing SAT 2017 (Cham, 2017), S. Gaspers and T. Walsh, Eds., Springer International Publishing, pp Raz, R., and Tzameret, I. Resolution over linear equations and multilinear proofs. Annals of Pure and Applied Logic 155, 3 (2008), Soos, M., Nohl, K., and Castelluccia, C. Extending SAT solvers to cryptographic problems. In International Conference on Theory and Applications of Satisfiability Testing (2009), Springer, pp Urquhart, A. Hard examples for resolution. J. ACM 34, 1 (Jan. 1987),
On Conversions from CNF to ANF
On Conversions from CNF to ANF Jan Horáček Martin Kreuzer Faculty of Informatics and Mathematics University of Passau, Germany Jan.Horacek@uni-passau.de Martin.Kreuzer@uni-passau.de Background ANF is XOR
More informationRESOLUTION OVER LINEAR EQUATIONS AND MULTILINEAR PROOFS
RESOLUTION OVER LINEAR EQUATIONS AND MULTILINEAR PROOFS RAN RAZ AND IDDO TZAMERET Abstract. We develop and study the complexity of propositional proof systems of varying strength extending resolution by
More informationPart 1: Propositional Logic
Part 1: Propositional Logic Literature (also for first-order logic) Schöning: Logik für Informatiker, Spektrum Fitting: First-Order Logic and Automated Theorem Proving, Springer 1 Last time 1.1 Syntax
More informationOn Conversions from CNF to ANF
On Conversions from CNF to ANF Jan Horáček and Martin Kreuzer Faculty of Informatics and Mathematics University of Passau, D-94030 Passau, Germany Jan.Horacek@uni-passau.de, Martin.Kreuzer@uni-passau.de
More informationALGEBRAIC PROOFS OVER NONCOMMUTATIVE FORMULAS
Electronic Colloquium on Computational Complexity, Report No. 97 (2010) ALGEBRAIC PROOFS OVER NONCOMMUTATIVE FORMULAS IDDO TZAMERET Abstract. We study possible formulations of algebraic propositional proof
More informationAdvanced Topics in LP and FP
Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection
More informationThe Strength of Multilinear Proofs
The Strength of Multilinear Proofs Ran Raz Iddo Tzameret December 19, 2006 Abstract We introduce an algebraic proof system that manipulates multilinear arithmetic formulas. We show that this proof system
More informationWarm-Up Problem. Is the following true or false? 1/35
Warm-Up Problem Is the following true or false? 1/35 Propositional Logic: Resolution Carmen Bruni Lecture 6 Based on work by J Buss, A Gao, L Kari, A Lubiw, B Bonakdarpour, D Maftuleac, C Roberts, R Trefler,
More informationPropositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) You will be expected to know Basic definitions Inference, derive, sound, complete Conjunctive Normal Form (CNF) Convert a Boolean formula to CNF Do a short
More informationLecture 11: Measuring the Complexity of Proofs
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Advanced Course on Computational Complexity Lecture 11: Measuring the Complexity of Proofs David Mix Barrington and Alexis Maciel July
More informationA New 3-CNF Transformation by Parallel-Serial Graphs 1
A New 3-CNF Transformation by Parallel-Serial Graphs 1 Uwe Bubeck, Hans Kleine Büning University of Paderborn, Computer Science Institute, 33098 Paderborn, Germany Abstract For propositional formulas we
More informationAlgebraic Proof Systems
Algebraic Proof Systems Pavel Pudlák Mathematical Institute, Academy of Sciences, Prague and Charles University, Prague Fall School of Logic, Prague, 2009 2 Overview 1 a survey of proof systems 2 a lower
More informationarxiv: v1 [cs.cc] 10 Aug 2007
RESOLUTION OVER LINEAR EQUATIONS AND MULTILINEAR PROOFS RAN RAZ AND IDDO TZAMERET arxiv:0708.1529v1 [cs.cc] 10 Aug 2007 Abstract. We develop and study the complexity of propositional proof systems of varying
More informationTitle: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5)
B.Y. Choueiry 1 Instructor s notes #12 Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5) Introduction to Artificial Intelligence CSCE 476-876, Fall 2018 URL: www.cse.unl.edu/ choueiry/f18-476-876
More informationApplied Logic. Lecture 1 - Propositional logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw
Applied Logic Lecture 1 - Propositional logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018
More informationPropositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
More informationLOGIC PROPOSITIONAL REASONING
LOGIC PROPOSITIONAL REASONING WS 2017/2018 (342.208) Armin Biere Martina Seidl biere@jku.at martina.seidl@jku.at Institute for Formal Models and Verification Johannes Kepler Universität Linz Version 2018.1
More informationChapter 7 R&N ICS 271 Fall 2017 Kalev Kask
Set 6: Knowledge Representation: The Propositional Calculus Chapter 7 R&N ICS 271 Fall 2017 Kalev Kask Outline Representing knowledge using logic Agent that reason logically A knowledge based agent Representing
More information7. Propositional Logic. Wolfram Burgard and Bernhard Nebel
Foundations of AI 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard and Bernhard Nebel Contents Agents that think rationally The wumpus world Propositional logic: syntax and semantics
More informationResolution and the Weak Pigeonhole Principle
Resolution and the Weak Pigeonhole Principle Sam Buss 1 and Toniann Pitassi 2 1 Departments of Mathematics and Computer Science University of California, San Diego, La Jolla, CA 92093-0112. 2 Department
More informationPropositional and Predicate Logic - V
Propositional and Predicate Logic - V Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - V WS 2016/2017 1 / 21 Formal proof systems Hilbert s calculus
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard, Maren Bennewitz, and Marco Ragni Albert-Ludwigs-Universität Freiburg Contents 1 Agents
More informationTopics in Model-Based Reasoning
Towards Integration of Proving and Solving Dipartimento di Informatica Università degli Studi di Verona Verona, Italy March, 2014 Automated reasoning Artificial Intelligence Automated Reasoning Computational
More informationSAT Solvers: Theory and Practice
Summer School on Verification Technology, Systems & Applications, September 17, 2008 p. 1/98 SAT Solvers: Theory and Practice Clark Barrett barrett@cs.nyu.edu New York University Summer School on Verification
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Joschka Boedecker and Wolfram Burgard and Bernhard Nebel Albert-Ludwigs-Universität Freiburg May 17, 2016
More informationPropositional Logic: Evaluating the Formulas
Institute for Formal Models and Verification Johannes Kepler University Linz VL Logik (LVA-Nr. 342208) Winter Semester 2015/2016 Propositional Logic: Evaluating the Formulas Version 2015.2 Armin Biere
More informationAn Introduction to SAT Solving
An Introduction to SAT Solving Applied Logic for Computer Science UWO December 3, 2017 Applied Logic for Computer Science An Introduction to SAT Solving UWO December 3, 2017 1 / 46 Plan 1 The Boolean satisfiability
More informationAutomated Program Verification and Testing 15414/15614 Fall 2016 Lecture 3: Practical SAT Solving
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 3: Practical SAT Solving Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson SAT Solving 1 / 36 Review: Propositional
More informationCritical Reading of Optimization Methods for Logical Inference [1]
Critical Reading of Optimization Methods for Logical Inference [1] Undergraduate Research Internship Department of Management Sciences Fall 2007 Supervisor: Dr. Miguel Anjos UNIVERSITY OF WATERLOO Rajesh
More informationAI Programming CS S-09 Knowledge Representation
AI Programming CS662-2013S-09 Knowledge Representation David Galles Department of Computer Science University of San Francisco 09-0: Overview So far, we ve talked about search, which is a means of considering
More informationYet Another Proof of the Strong Equivalence Between Propositional Theories and Logic Programs
Yet Another Proof of the Strong Equivalence Between Propositional Theories and Logic Programs Joohyung Lee and Ravi Palla School of Computing and Informatics Arizona State University, Tempe, AZ, USA {joolee,
More informationPropositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) This lecture topic: Propositional Logic (two lectures) Chapter 7.1-7.4 (previous lecture, Part I) Chapter 7.5 (this lecture, Part II) (optional: 7.6-7.8)
More informationPROOF COMPLEXITY IN ALGEBRAIC SYSTEMS AND BOUNDED DEPTH FREGE SYSTEMS WITH MODULAR COUNTING
PROOF COMPLEXITY IN ALGEBRAIC SYSTEMS AND BOUNDED DEPTH FREGE SYSTEMS WITH MODULAR COUNTING S. Buss, R. Impagliazzo, J. Krajíček, P. Pudlák, A. A. Razborov and J. Sgall Abstract. We prove a lower bound
More informationPrice: $25 (incl. T-Shirt, morning tea and lunch) Visit:
Three days of interesting talks & workshops from industry experts across Australia Explore new computing topics Network with students & employers in Brisbane Price: $25 (incl. T-Shirt, morning tea and
More informationComplexity of propositional proofs: Some theory and examples
Complexity of propositional proofs: Some theory and examples Univ. of California, San Diego Barcelona April 27, 2015 Frege proofs Introduction Frege proofs Pigeonhole principle Frege proofs are the usual
More informationKnowledge base (KB) = set of sentences in a formal language Declarative approach to building an agent (or other system):
Logic Knowledge-based agents Inference engine Knowledge base Domain-independent algorithms Domain-specific content Knowledge base (KB) = set of sentences in a formal language Declarative approach to building
More informationA brief introduction to Logic. (slides from
A brief introduction to Logic (slides from http://www.decision-procedures.org/) 1 A Brief Introduction to Logic - Outline Propositional Logic :Syntax Propositional Logic :Semantics Satisfiability and validity
More informationPropositional Logic: Methods of Proof. Chapter 7, Part II
Propositional Logic: Methods of Proof Chapter 7, Part II Inference in Formal Symbol Systems: Ontology, Representation, ti Inference Formal Symbol Systems Symbols correspond to things/ideas in the world
More information3 Propositional Logic
3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists
More informationLecture Notes on SAT Solvers & DPLL
15-414: Bug Catching: Automated Program Verification Lecture Notes on SAT Solvers & DPLL Matt Fredrikson André Platzer Carnegie Mellon University Lecture 10 1 Introduction In this lecture we will switch
More informationIntroduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
More information02 Propositional Logic
SE 2F03 Fall 2005 02 Propositional Logic Instructor: W. M. Farmer Revised: 25 September 2005 1 What is Propositional Logic? Propositional logic is the study of the truth or falsehood of propositions or
More informationOn the Structure and Complexity of Symbolic Proofs of Polynomial Identities
On the Structure and Complexity of Symbolic Proofs of Polynomial Identities Iddo Tzameret Tel Aviv University, Tel Aviv 69978, Israel Abstract A symbolic proof for establishing that a given arithmetic
More informationOrganization. Informal introduction and Overview Informal introductions to P,NP,co-NP and themes from and relationships with Proof complexity
15-16/08/2009 Nicola Galesi 1 Organization Informal introduction and Overview Informal introductions to P,NP,co-NP and themes from and relationships with Proof complexity First Steps in Proof Complexity
More informationLogic in AI Chapter 7. Mausam (Based on slides of Dan Weld, Stuart Russell, Dieter Fox, Henry Kautz )
Logic in AI Chapter 7 Mausam (Based on slides of Dan Weld, Stuart Russell, Dieter Fox, Henry Kautz ) Knowledge Representation represent knowledge in a manner that facilitates inferencing (i.e. drawing
More informationChapter 3: Propositional Calculus: Deductive Systems. September 19, 2008
Chapter 3: Propositional Calculus: Deductive Systems September 19, 2008 Outline 1 3.1 Deductive (Proof) System 2 3.2 Gentzen System G 3 3.3 Hilbert System H 4 3.4 Soundness and Completeness; Consistency
More informationClause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas
Journal of Artificial Intelligence Research 1 (1993) 1-15 Submitted 6/91; published 9/91 Clause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas Enrico Giunchiglia Massimo
More informationOn the Automatizability of Resolution and Related Propositional Proof Systems
On the Automatizability of Resolution and Related Propositional Proof Systems Albert Atserias and María Luisa Bonet Departament de Llenguatges i Sistemes Informàtics Universitat Politècnica de Catalunya
More informationPropositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173
Propositional Logic CSC 17 Propositional logic mathematical model (or algebra) for reasoning about the truth of logical expressions (propositions) Logical expressions propositional variables or logical
More informationVersion January Please send comments and corrections to
Mathematical Logic for Computer Science Second revised edition, Springer-Verlag London, 2001 Answers to Exercises Mordechai Ben-Ari Department of Science Teaching Weizmann Institute of Science Rehovot
More information6. Logical Inference
Artificial Intelligence 6. Logical Inference Prof. Bojana Dalbelo Bašić Assoc. Prof. Jan Šnajder University of Zagreb Faculty of Electrical Engineering and Computing Academic Year 2016/2017 Creative Commons
More informationRewriting for Satisfiability Modulo Theories
1 Dipartimento di Informatica Università degli Studi di Verona Verona, Italy July 10, 2010 1 Joint work with Chris Lynch (Department of Mathematics and Computer Science, Clarkson University, NY, USA) and
More informationClassical Propositional Logic
Classical Propositional Logic Peter Baumgartner http://users.cecs.anu.edu.au/~baumgart/ Ph: 02 6218 3717 Data61/CSIRO and ANU July 2017 1 / 71 Classical Logic and Reasoning Problems A 1 : Socrates is a
More information1 Algebraic Methods. 1.1 Gröbner Bases Applied to SAT
1 Algebraic Methods In an algebraic system Boolean constraints are expressed as a system of algebraic equations or inequalities which has a solution if and only if the constraints are satisfiable. Equations
More informationComputational Logic. Davide Martinenghi. Spring Free University of Bozen-Bolzano. Computational Logic Davide Martinenghi (1/30)
Computational Logic Davide Martinenghi Free University of Bozen-Bolzano Spring 2010 Computational Logic Davide Martinenghi (1/30) Propositional Logic - sequent calculus To overcome the problems of natural
More informationPythagorean Triples and SAT Solving
Pythagorean Triples and SAT Solving Moti Ben-Ari Department of Science Teaching Weizmann Institute of Science http://www.weizmann.ac.il/sci-tea/benari/ c 2017-18 by Moti Ben-Ari. This work is licensed
More informationLogic in AI Chapter 7. Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz )
Logic in AI Chapter 7 Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz ) 2 Knowledge Representation represent knowledge about the world in a manner that
More informationSupplementary Notes on Inductive Definitions
Supplementary Notes on Inductive Definitions 15-312: Foundations of Programming Languages Frank Pfenning Lecture 2 August 29, 2002 These supplementary notes review the notion of an inductive definition
More informationThe Wumpus Game. Stench Gold. Start. Cao Hoang Tru CSE Faculty - HCMUT
The Wumpus Game Stench Stench Gold Stench Start 1 The Wumpus Game Stench in the square containing the wumpus and in the directly adjacent squares in the squares directly adjacent to a pit Glitter in the
More informationPropositional logic (revision) & semantic entailment. p. 1/34
Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)
More informationGood Degree Bounds on Nullstellensatz Refutations of the Induction Principle
Good Degree Bounds on Nullstellensatz Refutations of the Induction Principle Samuel R. Buss Department of Mathematics University of California, San Diego Toniann Pitassi Department of Computer Science
More informationThe Deduction Theorem for Strong Propositional Proof Systems
The Deduction Theorem for Strong Propositional Proof Systems Olaf Beyersdorff Institut für Theoretische Informatik, Leibniz Universität Hannover, Germany beyersdorff@thi.uni-hannover.de Abstract. This
More informationEssential facts about NP-completeness:
CMPSCI611: NP Completeness Lecture 17 Essential facts about NP-completeness: Any NP-complete problem can be solved by a simple, but exponentially slow algorithm. We don t have polynomial-time solutions
More informationLecture 2: Symbolic Model Checking With SAT
Lecture 2: Symbolic Model Checking With SAT Edmund M. Clarke, Jr. School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 (Joint work over several years with: A. Biere, A. Cimatti, Y.
More informationArtificial Intelligence Chapter 7: Logical Agents
Artificial Intelligence Chapter 7: Logical Agents Michael Scherger Department of Computer Science Kent State University February 20, 2006 AI: Chapter 7: Logical Agents 1 Contents Knowledge Based Agents
More informationLogical Agents (I) Instructor: Tsung-Che Chiang
Logical Agents (I) Instructor: Tsung-Che Chiang tcchiang@ieee.org Department of Computer Science and Information Engineering National Taiwan Normal University Artificial Intelligence, Spring, 2010 編譯有誤
More informationarxiv: v2 [cs.ds] 17 Sep 2017
Two-Dimensional Indirect Binary Search for the Positive One-In-Three Satisfiability Problem arxiv:1708.08377v [cs.ds] 17 Sep 017 Shunichi Matsubara Aoyama Gakuin University, 5-10-1, Fuchinobe, Chuo-ku,
More informationDeliberative Agents Knowledge Representation I. Deliberative Agents
Deliberative Agents Knowledge Representation I Vasant Honavar Bioinformatics and Computational Biology Program Center for Computational Intelligence, Learning, & Discovery honavar@cs.iastate.edu www.cs.iastate.edu/~honavar/
More information7 LOGICAL AGENTS. OHJ-2556 Artificial Intelligence, Spring OHJ-2556 Artificial Intelligence, Spring
109 7 LOGICAL AGENS We now turn to knowledge-based agents that have a knowledge base KB at their disposal With the help of the KB the agent aims at maintaining knowledge of its partially-observable environment
More informationProof Complexity of Quantified Boolean Formulas
Proof Complexity of Quantified Boolean Formulas Olaf Beyersdorff School of Computing, University of Leeds Olaf Beyersdorff Proof Complexity of Quantified Boolean Formulas 1 / 39 Proof complexity (in one
More informationInference in Propositional Logic
Inference in Propositional Logic Deepak Kumar November 2017 Propositional Logic A language for symbolic reasoning Proposition a statement that is either True or False. E.g. Bryn Mawr College is located
More informationCS 771 Artificial Intelligence. Propositional Logic
CS 771 Artificial Intelligence Propositional Logic Why Do We Need Logic? Problem-solving agents were very inflexible hard code every possible state E.g., in the transition of 8-puzzle problem, knowledge
More informationPropositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) You will be expected to know Basic definitions Inference, derive, sound, complete Conjunctive Normal Form (CNF) Convert a Boolean formula to CNF Do a short
More informationCOMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning
COMP9414, Monday 26 March, 2012 Propositional Logic 2 COMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning Overview Proof systems (including soundness and completeness) Normal Forms
More informationA MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ
A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In
More informationPart 1: Propositional Logic
Part 1: Propositional Logic Literature (also for first-order logic) Schöning: Logik für Informatiker, Spektrum Fitting: First-Order Logic and Automated Theorem Proving, Springer 1 Last time 1.1 Syntax
More informationKnowledge representation DATA INFORMATION KNOWLEDGE WISDOM. Figure Relation ship between data, information knowledge and wisdom.
Knowledge representation Introduction Knowledge is the progression that starts with data which s limited utility. Data when processed become information, information when interpreted or evaluated becomes
More informationCardinality Networks: a Theoretical and Empirical Study
Constraints manuscript No. (will be inserted by the editor) Cardinality Networks: a Theoretical and Empirical Study Roberto Asín, Robert Nieuwenhuis, Albert Oliveras, Enric Rodríguez-Carbonell Received:
More informationPropositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
More informationRegular Resolution Lower Bounds for the Weak Pigeonhole Principle
Regular Resolution Lower Bounds for the Weak Pigeonhole Principle Toniann Pitassi Department of Computer Science University of Toronto toni@cs.toronto.edu Ran Raz Department of Computer Science Weizmann
More informationAlgebraic Proof Complexity: Progress, Frontiers and Challenges
Algebraic Proof Complexity: Progress, Frontiers and Challenges Toniann Pitassi Iddo Tzameret July 1, 2016 Abstract We survey recent progress in the proof complexity of strong proof systems and its connection
More informationA Generalized Method for Proving Polynomial Calculus Degree Lower Bounds
A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds Mladen Mikša and Jakob Nordström KTH Royal Institute of Technology SE-100 44 Stockholm, Sweden Abstract We study the problem of
More informationCS156: The Calculus of Computation
CS156: The Calculus of Computation Zohar Manna Winter 2010 It is reasonable to hope that the relationship between computation and mathematical logic will be as fruitful in the next century as that between
More informationINSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. A trade-off between length and width in resolution. Neil Thapen
INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES A trade-off between length and width in resolution Neil Thapen Preprint No. 59-2015 PRAHA 2015 A trade-off between length and width in resolution
More informationOn the Alekhnovich Razborov degree lower bound
On the Alekhnovich Razborov degree lower bound Yuval Filmus October 17, 2014 1 Introduction Polynomial calculus is a Hilbert-style proof system in which lines are polynomials modulo x 2 = x (for each variable
More informationLogical Closure Properties of Propositional Proof Systems
Logical Closure Properties of Propositional Proof Systems (Extended Abstract) Olaf Beyersdorff Institut für Theoretische Informatik, Leibniz Universität Hannover, Germany beyersdorff@thi.uni-hannover.de
More informationFirst-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms
First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO
More informationLinear Algebra, Boolean Rings and Resolution? Armin Biere. Institute for Formal Models and Verification Johannes Kepler University Linz, Austria
Linear Algebra, Boolean Rings and Resolution? Armin Biere Institute for Formal Models and Verification Johannes Kepler University Linz, Austria ACA 08 Applications of Computer Algebra Symbolic Computation
More informationPropositional Logic. Logic. Propositional Logic Syntax. Propositional Logic
Propositional Logic Reading: Chapter 7.1, 7.3 7.5 [ased on slides from Jerry Zhu, Louis Oliphant and ndrew Moore] Logic If the rules of the world are presented formally, then a decision maker can use logical
More informationInterleaved Alldifferent Constraints: CSP vs. SAT Approaches
Interleaved Alldifferent Constraints: CSP vs. SAT Approaches Frédéric Lardeux 3, Eric Monfroy 1,2, and Frédéric Saubion 3 1 Universidad Técnica Federico Santa María, Valparaíso, Chile 2 LINA, Université
More informationCS 4700: Foundations of Artificial Intelligence
CS 4700: Foundations of Artificial Intelligence Bart Selman selman@cs.cornell.edu Module: Knowledge, Reasoning, and Planning Part 2 Logical Agents R&N: Chapter 7 1 Illustrative example: Wumpus World (Somewhat
More informationLecture Notes on Inductive Definitions
Lecture Notes on Inductive Definitions 15-312: Foundations of Programming Languages Frank Pfenning Lecture 2 August 28, 2003 These supplementary notes review the notion of an inductive definition and give
More informationPropositional and Predicate Logic. jean/gbooks/logic.html
CMSC 630 February 10, 2009 1 Propositional and Predicate Logic Sources J. Gallier. Logic for Computer Science, John Wiley and Sons, Hoboken NJ, 1986. 2003 revised edition available on line at http://www.cis.upenn.edu/
More informationAttacking AES via SAT
Computer Science Department Swansea University BCTCS Warwick, April 7, 2009 Introduction In the following talk, a general translation framework, based around SAT, is considered, with the aim of providing
More informationOn evaluating decision procedures for modal logic
On evaluating decision procedures for modal logic Ullrich Hustadt and Renate A. Schmidt Max-Planck-Institut fur Informatik, 66123 Saarbriicken, Germany {hustadt, schmidt} topi-sb.mpg.de Abstract This paper
More informationFormal Verification Methods 1: Propositional Logic
Formal Verification Methods 1: Propositional Logic John Harrison Intel Corporation Course overview Propositional logic A resurgence of interest Logic and circuits Normal forms The Davis-Putnam procedure
More informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.
More informationSTUDIES IN ALGEBRAIC AND PROPOSITIONAL PROOF COMPLEXITY. Iddo Tzameret
TEL AVIV UNIVERSITY THE RAYMOND AND BEVERLY SACKLER FACULTY OF EXACT SCIENCES SCHOOL OF COMPUTER SCIENCE STUDIES IN ALGEBRAIC AND PROPOSITIONAL PROOF COMPLEXITY Thesis Submitted for the Degree of Doctor
More informationSatisfiability Modulo Theories
Satisfiability Modulo Theories Summer School on Formal Methods Menlo College, 2011 Bruno Dutertre and Leonardo de Moura bruno@csl.sri.com, leonardo@microsoft.com SRI International, Microsoft Research SAT/SMT
More informationLogical Agent & Propositional Logic
Logical Agent & Propositional Logic Berlin Chen 2005 References: 1. S. Russell and P. Norvig. Artificial Intelligence: A Modern Approach. Chapter 7 2. S. Russell s teaching materials Introduction The representation
More information